Computation of Green’s Function of the Bounded Solutions Problem

Abstract

Abstract It is well known that the equation x ′ ⁢ ( t ) = A ⁢ x ⁢ ( t ) + f ⁢ ( t ) {x^{\prime}(t)=Ax(t)+f(t)} , where A is a square matrix, has a unique bounded solution x for any bounded continuous free term f, provided the coefficient A has no eigenvalues on the imaginary axis. This solution can be represented in the form x ⁢ ( t ) = ∫ - ∞ ∞ 𝒢 ⁢ ( t - s ) ⁢ f ⁢ ( s ) ⁢ 𝑑 s . x(t)=\int_{-\infty}^{\infty}\mathcal{G}(t-s)f(s)\,ds. The kernel 𝒢 {\mathcal{G}} is called Green’s function. In this paper, for approximate calculation of 𝒢 {\mathcal{G}} , the Newton interpolating polynomial of a special function g t {g_{t}} is used. An estimate of the sensitivity of the problem is given. The results of numerical experiments are presented.